LMFDB - Embedded newform 140.10.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,10,Mod(1,140)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("140.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	140.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.1050170629$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9593x^{2} - 193749x - 162558 $ x^4 - x^3 - 9593x^2 - 193749x - 162558
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$107.329$ of defining polynomial
Character	$\chi$	$=$	140.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-149.145 q^{3} -625.000 q^{5} +2401.00 q^{7} +2561.29 q^{9} +O(q^{10})$	$q-149.145 q^{3} -625.000 q^{5} +2401.00 q^{7} +2561.29 q^{9} -75553.4 q^{11} +168258. q^{13} +93215.8 q^{15} +300889. q^{17} -539060. q^{19} -358098. q^{21} -33087.1 q^{23} +390625. q^{25} +2.55362e6 q^{27} +4.95041e6 q^{29} -296623. q^{31} +1.12684e7 q^{33} -1.50062e6 q^{35} +1.72962e7 q^{37} -2.50949e7 q^{39} -1.44256e7 q^{41} -1.63386e6 q^{43} -1.60081e6 q^{45} +3.49642e7 q^{47} +5.76480e6 q^{49} -4.48761e7 q^{51} -6.73746e7 q^{53} +4.72209e7 q^{55} +8.03983e7 q^{57} +1.05066e8 q^{59} -1.24809e8 q^{61} +6.14967e6 q^{63} -1.05162e8 q^{65} -3.10101e8 q^{67} +4.93479e6 q^{69} +3.53271e7 q^{71} -2.25107e8 q^{73} -5.82598e7 q^{75} -1.81404e8 q^{77} -4.88743e8 q^{79} -4.31274e8 q^{81} +6.43113e8 q^{83} -1.88056e8 q^{85} -7.38330e8 q^{87} +8.83597e7 q^{89} +4.03989e8 q^{91} +4.42399e7 q^{93} +3.36913e8 q^{95} -1.30157e9 q^{97} -1.93514e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 43 q^{3} - 2500 q^{5} + 9604 q^{7} + 26935 q^{9}+O(q^{10}) $ 4 * q + 43 * q^3 - 2500 * q^5 + 9604 * q^7 + 26935 * q^9	$ 4 q + 43 q^{3} - 2500 q^{5} + 9604 q^{7} + 26935 q^{9} - 89509 q^{11} - 26951 q^{13} - 26875 q^{15} - 97685 q^{17} + 57614 q^{19} + 103243 q^{21} + 704846 q^{23} + 1562500 q^{25} + 3392209 q^{27} + 6396351 q^{29} - 2177396 q^{31} - 1814697 q^{33} - 6002500 q^{35} - 4832420 q^{37} - 20266753 q^{39} - 33404930 q^{41} - 23422170 q^{43} - 16834375 q^{45} - 30030109 q^{47} + 23059204 q^{49} - 101036823 q^{51} - 29121822 q^{53} + 55943125 q^{55} + 46465094 q^{57} + 187841888 q^{59} - 320985014 q^{61} + 64670935 q^{63} + 16844375 q^{65} - 257915784 q^{67} - 342824850 q^{69} + 95380960 q^{71} - 274354616 q^{73} + 16796875 q^{75} - 214911109 q^{77} - 369481275 q^{79} - 1415449364 q^{81} - 87594732 q^{83} + 61053125 q^{85} - 971702439 q^{87} - 1556249690 q^{89} - 64709351 q^{91} - 1709587660 q^{93} - 36008750 q^{95} - 3692761793 q^{97} - 1570184394 q^{99}+O(q^{100}) $ 4 * q + 43 * q^3 - 2500 * q^5 + 9604 * q^7 + 26935 * q^9 - 89509 * q^11 - 26951 * q^13 - 26875 * q^15 - 97685 * q^17 + 57614 * q^19 + 103243 * q^21 + 704846 * q^23 + 1562500 * q^25 + 3392209 * q^27 + 6396351 * q^29 - 2177396 * q^31 - 1814697 * q^33 - 6002500 * q^35 - 4832420 * q^37 - 20266753 * q^39 - 33404930 * q^41 - 23422170 * q^43 - 16834375 * q^45 - 30030109 * q^47 + 23059204 * q^49 - 101036823 * q^51 - 29121822 * q^53 + 55943125 * q^55 + 46465094 * q^57 + 187841888 * q^59 - 320985014 * q^61 + 64670935 * q^63 + 16844375 * q^65 - 257915784 * q^67 - 342824850 * q^69 + 95380960 * q^71 - 274354616 * q^73 + 16796875 * q^75 - 214911109 * q^77 - 369481275 * q^79 - 1415449364 * q^81 - 87594732 * q^83 + 61053125 * q^85 - 971702439 * q^87 - 1556249690 * q^89 - 64709351 * q^91 - 1709587660 * q^93 - 36008750 * q^95 - 3692761793 * q^97 - 1570184394 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−149.145	−1.06307	−0.531537	−	0.847035i	$-0.678385\pi$
$3$	−149.145	−1.06307	−0.531537	+	0.847035i	$0.678385\pi$
$4$	0	0
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	0	0
$9$	2561.29	0.130127
$10$	0	0
$11$	−75553.4	−1.55592	−0.777959	−	0.628315i	$-0.783745\pi$
$11$	−75553.4	−1.55592	−0.777959	+	0.628315i	$0.783745\pi$
$12$	0	0
$13$	168258.	1.63392	0.816962	−	0.576691i	$-0.195656\pi$
$13$	168258.	1.63392	0.816962	+	0.576691i	$0.195656\pi$
$14$	0	0
$15$	93215.8	0.475421
$16$	0	0
$17$	300889.	0.873748	0.436874	−	0.899523i	$-0.356086\pi$
$17$	300889.	0.873748	0.436874	+	0.899523i	$0.356086\pi$
$18$	0	0
$19$	−539060.	−0.948956	−0.474478	−	0.880267i	$-0.657363\pi$
$19$	−539060.	−0.948956	−0.474478	+	0.880267i	$0.657363\pi$
$20$	0	0
$21$	−358098.	−0.401804
$22$	0	0
$23$	−33087.1	−0.0246538	−0.0123269	−	0.999924i	$-0.503924\pi$
$23$	−33087.1	−0.0246538	−0.0123269	+	0.999924i	$0.503924\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	2.55362e6	0.924740
$28$	0	0
$29$	4.95041e6	1.29972	0.649861	−	0.760053i	$-0.274827\pi$
$29$	4.95041e6	1.29972	0.649861	+	0.760053i	$0.274827\pi$
$30$	0	0
$31$	−296623.	−0.0576869	−0.0288434	−	0.999584i	$-0.509182\pi$
$31$	−296623.	−0.0576869	−0.0288434	+	0.999584i	$0.509182\pi$
$32$	0	0
$33$	1.12684e7	1.65406
$34$	0	0
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	1.72962e7	1.51720	0.758601	−	0.651556i	$-0.225883\pi$
$37$	1.72962e7	1.51720	0.758601	+	0.651556i	$0.225883\pi$
$38$	0	0
$39$	−2.50949e7	−1.73698
$40$	0	0
$41$	−1.44256e7	−0.797272	−0.398636	−	0.917109i	$-0.630516\pi$
$41$	−1.44256e7	−0.797272	−0.398636	+	0.917109i	$0.630516\pi$
$42$	0	0
$43$	−1.63386e6	−0.0728797	−0.0364398	−	0.999336i	$-0.511602\pi$
$43$	−1.63386e6	−0.0728797	−0.0364398	+	0.999336i	$0.511602\pi$
$44$	0	0
$45$	−1.60081e6	−0.0581947
$46$	0	0
$47$	3.49642e7	1.04516	0.522581	−	0.852589i	$-0.324969\pi$
$47$	3.49642e7	1.04516	0.522581	+	0.852589i	$0.324969\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	−4.48761e7	−0.928859
$52$	0	0
$53$	−6.73746e7	−1.17288	−0.586442	−	0.809991i	$-0.699472\pi$
$53$	−6.73746e7	−1.17288	−0.586442	+	0.809991i	$0.699472\pi$
$54$	0	0
$55$	4.72209e7	0.695828
$56$	0	0
$57$	8.03983e7	1.00881
$58$	0	0
$59$	1.05066e8	1.12883	0.564413	−	0.825493i	$-0.309103\pi$
$59$	1.05066e8	1.12883	0.564413	+	0.825493i	$0.309103\pi$
$60$	0	0
$61$	−1.24809e8	−1.15415	−0.577076	−	0.816691i	$-0.695806\pi$
$61$	−1.24809e8	−1.15415	−0.577076	+	0.816691i	$0.695806\pi$
$62$	0	0
$63$	6.14967e6	0.0491835
$64$	0	0
$65$	−1.05162e8	−0.730713
$66$	0	0
$67$	−3.10101e8	−1.88004	−0.940020	−	0.341121i	$-0.889194\pi$
$67$	−3.10101e8	−1.88004	−0.940020	+	0.341121i	$0.889194\pi$
$68$	0	0
$69$	4.93479e6	0.0262088
$70$	0	0
$71$	3.53271e7	0.164985	0.0824927	−	0.996592i	$-0.473712\pi$
$71$	3.53271e7	0.164985	0.0824927	+	0.996592i	$0.473712\pi$
$72$	0	0
$73$	−2.25107e8	−0.927761	−0.463881	−	0.885898i	$-0.653543\pi$
$73$	−2.25107e8	−0.927761	−0.463881	+	0.885898i	$0.653543\pi$
$74$	0	0
$75$	−5.82598e7	−0.212615
$76$	0	0
$77$	−1.81404e8	−0.588082
$78$	0	0
$79$	−4.88743e8	−1.41175	−0.705876	−	0.708335i	$-0.749447\pi$
$79$	−4.88743e8	−1.41175	−0.705876	+	0.708335i	$0.749447\pi$
$80$	0	0
$81$	−4.31274e8	−1.11319
$82$	0	0
$83$	6.43113e8	1.48743	0.743714	−	0.668498i	$-0.233063\pi$
$83$	6.43113e8	1.48743	0.743714	+	0.668498i	$0.233063\pi$
$84$	0	0
$85$	−1.88056e8	−0.390752
$86$	0	0
$87$	−7.38330e8	−1.38170
$88$	0	0
$89$	8.83597e7	0.149279	0.0746396	−	0.997211i	$-0.476219\pi$
$89$	8.83597e7	0.149279	0.0746396	+	0.997211i	$0.476219\pi$
$90$	0	0
$91$	4.03989e8	0.617565
$92$	0	0
$93$	4.42399e7	0.0613254
$94$	0	0
$95$	3.36913e8	0.424386
$96$	0	0
$97$	−1.30157e9	−1.49278	−0.746388	−	0.665511i	$-0.768213\pi$
$97$	−1.30157e9	−1.49278	−0.746388	+	0.665511i	$0.768213\pi$
$98$	0	0
$99$	−1.93514e8	−0.202467
$100$	0	0
$101$	6.26483e8	0.599050	0.299525	−	0.954088i	$-0.403172\pi$
$101$	6.26483e8	0.599050	0.299525	+	0.954088i	$0.403172\pi$
$102$	0	0
$103$	−1.97871e9	−1.73227	−0.866135	−	0.499810i	$-0.833403\pi$
$103$	−1.97871e9	−1.73227	−0.866135	+	0.499810i	$0.833403\pi$
$104$	0	0
$105$	2.23811e8	0.179692
$106$	0	0
$107$	−9.16274e8	−0.675769	−0.337885	−	0.941187i	$-0.609711\pi$
$107$	−9.16274e8	−0.675769	−0.337885	+	0.941187i	$0.609711\pi$
$108$	0	0
$109$	−2.33142e8	−0.158198	−0.0790992	−	0.996867i	$-0.525204\pi$
$109$	−2.33142e8	−0.158198	−0.0790992	+	0.996867i	$0.525204\pi$
$110$	0	0
$111$	−2.57965e9	−1.61290
$112$	0	0
$113$	9.68456e8	0.558762	0.279381	−	0.960180i	$-0.409871\pi$
$113$	9.68456e8	0.558762	0.279381	+	0.960180i	$0.409871\pi$
$114$	0	0
$115$	2.06794e7	0.0110255
$116$	0	0
$117$	4.30959e8	0.212618
$118$	0	0
$119$	7.22434e8	0.330246
$120$	0	0
$121$	3.35037e9	1.42088
$122$	0	0
$123$	2.15151e9	0.847560
$124$	0	0
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	−2.43643e9	−0.831068	−0.415534	−	0.909578i	$-0.636405\pi$
$127$	−2.43643e9	−0.831068	−0.415534	+	0.909578i	$0.636405\pi$
$128$	0	0
$129$	2.43682e8	0.0774765
$130$	0	0
$131$	−1.34218e9	−0.398190	−0.199095	−	0.979980i	$-0.563800\pi$
$131$	−1.34218e9	−0.398190	−0.199095	+	0.979980i	$0.563800\pi$
$132$	0	0
$133$	−1.29428e9	−0.358672
$134$	0	0
$135$	−1.59601e9	−0.413556
$136$	0	0
$137$	2.87982e9	0.698431	0.349215	−	0.937042i	$-0.386448\pi$
$137$	2.87982e9	0.698431	0.349215	+	0.937042i	$0.386448\pi$
$138$	0	0
$139$	−4.81982e9	−1.09513	−0.547563	−	0.836765i	$-0.684444\pi$
$139$	−4.81982e9	−1.09513	−0.547563	+	0.836765i	$0.684444\pi$
$140$	0	0
$141$	−5.21475e9	−1.11109
$142$	0	0
$143$	−1.27125e10	−2.54225
$144$	0	0
$145$	−3.09401e9	−0.581253
$146$	0	0
$147$	−8.59792e8	−0.151868
$148$	0	0
$149$	−1.35420e9	−0.225085	−0.112542	−	0.993647i	$-0.535899\pi$
$149$	−1.35420e9	−0.225085	−0.112542	+	0.993647i	$0.535899\pi$
$150$	0	0
$151$	−3.74271e9	−0.585854	−0.292927	−	0.956135i	$-0.594629\pi$
$151$	−3.74271e9	−0.585854	−0.292927	+	0.956135i	$0.594629\pi$
$152$	0	0
$153$	7.70665e8	0.113698
$154$	0	0
$155$	1.85389e8	0.0257983
$156$	0	0
$157$	−4.35779e9	−0.572424	−0.286212	−	0.958166i	$-0.592396\pi$
$157$	−4.35779e9	−0.572424	−0.286212	+	0.958166i	$0.592396\pi$
$158$	0	0
$159$	1.00486e10	1.24686
$160$	0	0
$161$	−7.94422e7	−0.00931826
$162$	0	0
$163$	9.66422e9	1.07232	0.536158	−	0.844118i	$-0.319875\pi$
$163$	9.66422e9	1.07232	0.536158	+	0.844118i	$0.319875\pi$
$164$	0	0
$165$	−7.04277e9	−0.739717
$166$	0	0
$167$	−1.01198e10	−1.00681	−0.503405	−	0.864051i	$-0.667919\pi$
$167$	−1.01198e10	−1.00681	−0.503405	+	0.864051i	$0.667919\pi$
$168$	0	0
$169$	1.77064e10	1.66971
$170$	0	0
$171$	−1.38069e9	−0.123485
$172$	0	0
$173$	8.43123e9	0.715621	0.357811	−	0.933794i	$-0.383523\pi$
$173$	8.43123e9	0.715621	0.357811	+	0.933794i	$0.383523\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	0	0
$177$	−1.56700e10	−1.20003
$178$	0	0
$179$	5.45528e9	0.397172	0.198586	−	0.980084i	$-0.436365\pi$
$179$	5.45528e9	0.397172	0.198586	+	0.980084i	$0.436365\pi$
$180$	0	0
$181$	2.56526e10	1.77655	0.888276	−	0.459310i	$-0.151903\pi$
$181$	2.56526e10	1.77655	0.888276	+	0.459310i	$0.151903\pi$
$182$	0	0
$183$	1.86147e10	1.22695
$184$	0	0
$185$	−1.08101e10	−0.678513
$186$	0	0
$187$	−2.27332e10	−1.35948
$188$	0	0
$189$	6.13124e9	0.349519
$190$	0	0
$191$	−2.62735e10	−1.42846	−0.714231	−	0.699910i	$-0.753223\pi$
$191$	−2.62735e10	−1.42846	−0.714231	+	0.699910i	$0.753223\pi$
$192$	0	0
$193$	1.05745e10	0.548597	0.274299	−	0.961645i	$-0.411554\pi$
$193$	1.05745e10	0.548597	0.274299	+	0.961645i	$0.411554\pi$
$194$	0	0
$195$	1.56843e10	0.776802
$196$	0	0
$197$	6.54384e9	0.309553	0.154776	−	0.987950i	$-0.450534\pi$
$197$	6.54384e9	0.309553	0.154776	+	0.987950i	$0.450534\pi$
$198$	0	0
$199$	9.16084e9	0.414092	0.207046	−	0.978331i	$-0.433615\pi$
$199$	9.16084e9	0.414092	0.207046	+	0.978331i	$0.433615\pi$
$200$	0	0
$201$	4.62501e10	1.99862
$202$	0	0
$203$	1.18859e10	0.491248
$204$	0	0
$205$	9.01600e9	0.356551
$206$	0	0
$207$	−8.47458e7	−0.00320813
$208$	0	0
$209$	4.07278e10	1.47650
$210$	0	0
$211$	2.70152e10	0.938291	0.469146	−	0.883121i	$-0.344562\pi$
$211$	2.70152e10	0.938291	0.469146	+	0.883121i	$0.344562\pi$
$212$	0	0
$213$	−5.26887e9	−0.175392
$214$	0	0
$215$	1.02116e9	0.0325928
$216$	0	0
$217$	−7.12191e8	−0.0218036
$218$	0	0
$219$	3.35736e10	0.986279
$220$	0	0
$221$	5.06271e10	1.42764
$222$	0	0
$223$	−4.51719e10	−1.22320	−0.611598	−	0.791169i	$-0.709473\pi$
$223$	−4.51719e10	−1.22320	−0.611598	+	0.791169i	$0.709473\pi$
$224$	0	0
$225$	1.00051e9	0.0260254
$226$	0	0
$227$	3.00064e9	0.0750062	0.0375031	−	0.999297i	$-0.488060\pi$
$227$	3.00064e9	0.0750062	0.0375031	+	0.999297i	$0.488060\pi$
$228$	0	0
$229$	−7.75100e10	−1.86251	−0.931254	−	0.364372i	$-0.881284\pi$
$229$	−7.75100e10	−1.86251	−0.931254	+	0.364372i	$0.881284\pi$
$230$	0	0
$231$	2.70555e10	0.625175
$232$	0	0
$233$	5.52066e10	1.22713	0.613564	−	0.789645i	$-0.289735\pi$
$233$	5.52066e10	1.22713	0.613564	+	0.789645i	$0.289735\pi$
$234$	0	0
$235$	−2.18527e10	−0.467411
$236$	0	0
$237$	7.28937e10	1.50080
$238$	0	0
$239$	7.52484e10	1.49179	0.745893	−	0.666065i	$-0.232023\pi$
$239$	7.52484e10	1.49179	0.745893	+	0.666065i	$0.232023\pi$
$240$	0	0
$241$	−5.60865e10	−1.07098	−0.535491	−	0.844541i	$-0.679873\pi$
$241$	−5.60865e10	−1.07098	−0.535491	+	0.844541i	$0.679873\pi$
$242$	0	0
$243$	1.40596e10	0.258669
$244$	0	0
$245$	−3.60300e9	−0.0638877
$246$	0	0
$247$	−9.07015e10	−1.55052
$248$	0	0
$249$	−9.59172e10	−1.58125
$250$	0	0
$251$	−6.37099e10	−1.01315	−0.506576	−	0.862195i	$-0.669089\pi$
$251$	−6.37099e10	−1.01315	−0.506576	+	0.862195i	$0.669089\pi$
$252$	0	0
$253$	2.49984e9	0.0383593
$254$	0	0
$255$	2.80476e10	0.415398
$256$	0	0
$257$	−7.72295e10	−1.10429	−0.552146	−	0.833747i	$-0.686191\pi$
$257$	−7.72295e10	−1.10429	−0.552146	+	0.833747i	$0.686191\pi$
$258$	0	0
$259$	4.15282e10	0.573448
$260$	0	0
$261$	1.26795e10	0.169129
$262$	0	0
$263$	1.21784e11	1.56960	0.784802	−	0.619746i	$-0.212764\pi$
$263$	1.21784e11	1.56960	0.784802	+	0.619746i	$0.212764\pi$
$264$	0	0
$265$	4.21092e10	0.524530
$266$	0	0
$267$	−1.31784e10	−0.158695
$268$	0	0
$269$	−4.05352e10	−0.472005	−0.236003	−	0.971752i	$-0.575837\pi$
$269$	−4.05352e10	−0.472005	−0.236003	+	0.971752i	$0.575837\pi$
$270$	0	0
$271$	2.51105e10	0.282809	0.141404	−	0.989952i	$-0.454838\pi$
$271$	2.51105e10	0.282809	0.141404	+	0.989952i	$0.454838\pi$
$272$	0	0
$273$	−6.02530e10	−0.656518
$274$	0	0
$275$	−2.95130e10	−0.311184
$276$	0	0
$277$	1.91842e11	1.95787	0.978935	−	0.204174i	$-0.0654507\pi$
$277$	1.91842e11	1.95787	0.978935	+	0.204174i	$0.0654507\pi$
$278$	0	0
$279$	−7.59738e8	−0.00750663
$280$	0	0
$281$	−1.23387e11	−1.18057	−0.590284	−	0.807196i	$-0.700984\pi$
$281$	−1.23387e11	−1.18057	−0.590284	+	0.807196i	$0.700984\pi$
$282$	0	0
$283$	−1.00503e11	−0.931406	−0.465703	−	0.884941i	$-0.654198\pi$
$283$	−1.00503e11	−0.931406	−0.465703	+	0.884941i	$0.654198\pi$
$284$	0	0
$285$	−5.02489e10	−0.451154
$286$	0	0
$287$	−3.46359e10	−0.301341
$288$	0	0
$289$	−2.80537e10	−0.236565
$290$	0	0
$291$	1.94123e11	1.58693
$292$	0	0
$293$	3.58007e10	0.283783	0.141892	−	0.989882i	$-0.454682\pi$
$293$	3.58007e10	0.283783	0.141892	+	0.989882i	$0.454682\pi$
$294$	0	0
$295$	−6.56661e10	−0.504826
$296$	0	0
$297$	−1.92935e11	−1.43882
$298$	0	0
$299$	−5.56719e9	−0.0402824
$300$	0	0
$301$	−3.92289e9	−0.0275459
$302$	0	0
$303$	−9.34370e10	−0.636835
$304$	0	0
$305$	7.80058e10	0.516152
$306$	0	0
$307$	9.14759e10	0.587738	0.293869	−	0.955846i	$-0.405057\pi$
$307$	9.14759e10	0.587738	0.293869	+	0.955846i	$0.405057\pi$
$308$	0	0
$309$	2.95116e11	1.84153
$310$	0	0
$311$	−2.68769e11	−1.62913	−0.814567	−	0.580070i	$-0.803025\pi$
$311$	−2.68769e11	−1.62913	−0.814567	+	0.580070i	$0.803025\pi$
$312$	0	0
$313$	−1.78146e10	−0.104912	−0.0524562	−	0.998623i	$-0.516705\pi$
$313$	−1.78146e10	−0.104912	−0.0524562	+	0.998623i	$0.516705\pi$
$314$	0	0
$315$	−3.84354e9	−0.0219955
$316$	0	0
$317$	−1.76263e11	−0.980381	−0.490191	−	0.871615i	$-0.663073\pi$
$317$	−1.76263e11	−0.980381	−0.490191	+	0.871615i	$0.663073\pi$
$318$	0	0
$319$	−3.74020e11	−2.02226
$320$	0	0
$321$	1.36658e11	0.718393
$322$	0	0
$323$	−1.62197e11	−0.829149
$324$	0	0
$325$	6.57260e10	0.326785
$326$	0	0
$327$	3.47721e10	0.168177
$328$	0	0
$329$	8.39492e10	0.395034
$330$	0	0
$331$	2.52623e11	1.15677	0.578384	−	0.815765i	$-0.303683\pi$
$331$	2.52623e11	1.15677	0.578384	+	0.815765i	$0.303683\pi$
$332$	0	0
$333$	4.43007e10	0.197429
$334$	0	0
$335$	1.93813e11	0.840779
$336$	0	0
$337$	5.02141e10	0.212076	0.106038	−	0.994362i	$-0.466184\pi$
$337$	5.02141e10	0.212076	0.106038	+	0.994362i	$0.466184\pi$
$338$	0	0
$339$	−1.44441e11	−0.594005
$340$	0	0
$341$	2.24109e10	0.0897561
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	0	0
$345$	−3.08424e9	−0.0117209
$346$	0	0
$347$	−2.81466e11	−1.04218	−0.521090	−	0.853502i	$-0.674474\pi$
$347$	−2.81466e11	−1.04218	−0.521090	+	0.853502i	$0.674474\pi$
$348$	0	0
$349$	−4.64545e11	−1.67615	−0.838075	−	0.545555i	$-0.816319\pi$
$349$	−4.64545e11	−1.67615	−0.838075	+	0.545555i	$0.816319\pi$
$350$	0	0
$351$	4.29668e11	1.51095
$352$	0	0
$353$	−5.31828e11	−1.82299	−0.911497	−	0.411306i	$-0.865073\pi$
$353$	−5.31828e11	−1.82299	−0.911497	+	0.411306i	$0.865073\pi$
$354$	0	0
$355$	−2.20794e10	−0.0737837
$356$	0	0
$357$	−1.07748e11	−0.351076
$358$	0	0
$359$	−5.82292e11	−1.85019	−0.925094	−	0.379739i	$-0.876014\pi$
$359$	−5.82292e11	−1.85019	−0.925094	+	0.379739i	$0.876014\pi$
$360$	0	0
$361$	−3.21015e10	−0.0994816
$362$	0	0
$363$	−4.99691e11	−1.51050
$364$	0	0
$365$	1.40692e11	0.414907
$366$	0	0
$367$	2.70055e11	0.777061	0.388530	−	0.921436i	$-0.372983\pi$
$367$	2.70055e11	0.777061	0.388530	+	0.921436i	$0.372983\pi$
$368$	0	0
$369$	−3.69482e10	−0.103747
$370$	0	0
$371$	−1.61767e11	−0.443308
$372$	0	0
$373$	1.68380e11	0.450403	0.225202	−	0.974312i	$-0.427696\pi$
$373$	1.68380e11	0.450403	0.225202	+	0.974312i	$0.427696\pi$
$374$	0	0
$375$	3.64124e10	0.0950843
$376$	0	0
$377$	8.32949e11	2.12365
$378$	0	0
$379$	−6.43073e11	−1.60097	−0.800487	−	0.599351i	$-0.795426\pi$
$379$	−6.43073e11	−1.60097	−0.800487	+	0.599351i	$0.795426\pi$
$380$	0	0
$381$	3.63381e11	0.883487
$382$	0	0
$383$	1.01450e10	0.0240912	0.0120456	−	0.999927i	$-0.496166\pi$
$383$	1.01450e10	0.0240912	0.0120456	+	0.999927i	$0.496166\pi$
$384$	0	0
$385$	1.13377e11	0.262998
$386$	0	0
$387$	−4.18479e9	−0.00948363
$388$	0	0
$389$	−6.61714e11	−1.46520	−0.732600	−	0.680659i	$-0.761693\pi$
$389$	−6.61714e11	−1.46520	−0.732600	+	0.680659i	$0.761693\pi$
$390$	0	0
$391$	−9.95555e9	−0.0215412
$392$	0	0
$393$	2.00180e11	0.423305
$394$	0	0
$395$	3.05464e11	0.631355
$396$	0	0
$397$	−6.34210e11	−1.28137	−0.640687	−	0.767802i	$-0.721351\pi$
$397$	−6.34210e11	−1.28137	−0.640687	+	0.767802i	$0.721351\pi$
$398$	0	0
$399$	1.93036e11	0.381295
$400$	0	0
$401$	5.45611e11	1.05374	0.526870	−	0.849946i	$-0.323365\pi$
$401$	5.45611e11	1.05374	0.526870	+	0.849946i	$0.323365\pi$
$402$	0	0
$403$	−4.99093e10	−0.0942560
$404$	0	0
$405$	2.69546e11	0.497836
$406$	0	0
$407$	−1.30679e12	−2.36064
$408$	0	0
$409$	2.85875e11	0.505152	0.252576	−	0.967577i	$-0.418722\pi$
$409$	2.85875e11	0.505152	0.252576	+	0.967577i	$0.418722\pi$
$410$	0	0
$411$	−4.29512e11	−0.742484
$412$	0	0
$413$	2.52263e11	0.426656
$414$	0	0
$415$	−4.01946e11	−0.665198
$416$	0	0
$417$	7.18852e11	1.16420
$418$	0	0
$419$	9.01112e10	0.142829	0.0714143	−	0.997447i	$-0.477249\pi$
$419$	9.01112e10	0.142829	0.0714143	+	0.997447i	$0.477249\pi$
$420$	0	0
$421$	3.07097e11	0.476437	0.238218	−	0.971212i	$-0.423437\pi$
$421$	3.07097e11	0.476437	0.238218	+	0.971212i	$0.423437\pi$
$422$	0	0
$423$	8.95537e10	0.136004
$424$	0	0
$425$	1.17535e11	0.174750
$426$	0	0
$427$	−2.99667e11	−0.436228
$428$	0	0
$429$	1.89601e12	2.70260
$430$	0	0
$431$	−7.33835e11	−1.02436	−0.512178	−	0.858879i	$-0.671161\pi$
$431$	−7.33835e11	−1.02436	−0.512178	+	0.858879i	$0.671161\pi$
$432$	0	0
$433$	−6.57786e11	−0.899268	−0.449634	−	0.893213i	$-0.648446\pi$
$433$	−6.57786e11	−0.899268	−0.449634	+	0.893213i	$0.648446\pi$
$434$	0	0
$435$	4.61456e11	0.617915
$436$	0	0
$437$	1.78360e10	0.0233954
$438$	0	0
$439$	−1.02525e12	−1.31746	−0.658731	−	0.752379i	$-0.728906\pi$
$439$	−1.02525e12	−1.31746	−0.658731	+	0.752379i	$0.728906\pi$
$440$	0	0
$441$	1.47653e10	0.0185896
$442$	0	0
$443$	1.02998e12	1.27061	0.635303	−	0.772263i	$-0.280875\pi$
$443$	1.02998e12	1.27061	0.635303	+	0.772263i	$0.280875\pi$
$444$	0	0
$445$	−5.52248e10	−0.0667597
$446$	0	0
$447$	2.01973e11	0.239282
$448$	0	0
$449$	1.14772e12	1.33268	0.666341	−	0.745647i	$-0.267860\pi$
$449$	1.14772e12	1.33268	0.666341	+	0.745647i	$0.267860\pi$
$450$	0	0
$451$	1.08990e12	1.24049
$452$	0	0
$453$	5.58207e11	0.622806
$454$	0	0
$455$	−2.52493e11	−0.276184
$456$	0	0
$457$	5.85582e11	0.628007	0.314004	−	0.949422i	$-0.398330\pi$
$457$	5.85582e11	0.628007	0.314004	+	0.949422i	$0.398330\pi$
$458$	0	0
$459$	7.68356e11	0.807989
$460$	0	0
$461$	1.17304e12	1.20965	0.604823	−	0.796360i	$-0.293244\pi$
$461$	1.17304e12	1.20965	0.604823	+	0.796360i	$0.293244\pi$
$462$	0	0
$463$	4.69481e11	0.474792	0.237396	−	0.971413i	$-0.423706\pi$
$463$	4.69481e11	0.474792	0.237396	+	0.971413i	$0.423706\pi$
$464$	0	0
$465$	−2.76499e10	−0.0274256
$466$	0	0
$467$	2.96739e10	0.0288702	0.0144351	−	0.999896i	$-0.495405\pi$
$467$	2.96739e10	0.0288702	0.0144351	+	0.999896i	$0.495405\pi$
$468$	0	0
$469$	−7.44553e11	−0.710588
$470$	0	0
$471$	6.49944e11	0.608530
$472$	0	0
$473$	1.23444e11	0.113395
$474$	0	0
$475$	−2.10571e11	−0.189791
$476$	0	0
$477$	−1.72566e11	−0.152624
$478$	0	0
$479$	−1.55383e12	−1.34863	−0.674315	−	0.738444i	$-0.735561\pi$
$479$	−1.55383e12	−1.34863	−0.674315	+	0.738444i	$0.735561\pi$
$480$	0	0
$481$	2.91023e12	2.47899
$482$	0	0
$483$	1.18484e10	0.00990600
$484$	0	0
$485$	8.13481e11	0.667589
$486$	0	0
$487$	−1.53759e12	−1.23868	−0.619341	−	0.785122i	$-0.712600\pi$
$487$	−1.53759e12	−1.23868	−0.619341	+	0.785122i	$0.712600\pi$
$488$	0	0
$489$	−1.44137e12	−1.13995
$490$	0	0
$491$	1.05265e12	0.817369	0.408684	−	0.912676i	$-0.365988\pi$
$491$	1.05265e12	0.817369	0.408684	+	0.912676i	$0.365988\pi$
$492$	0	0
$493$	1.48952e12	1.13563
$494$	0	0
$495$	1.20947e11	0.0905461
$496$	0	0
$497$	8.48204e10	0.0623586
$498$	0	0
$499$	1.84002e12	1.32852	0.664261	−	0.747500i	$-0.268746\pi$
$499$	1.84002e12	1.32852	0.664261	+	0.747500i	$0.268746\pi$
$500$	0	0
$501$	1.50932e12	1.07031
$502$	0	0
$503$	1.10802e12	0.771774	0.385887	−	0.922546i	$-0.373895\pi$
$503$	1.10802e12	0.771774	0.385887	+	0.922546i	$0.373895\pi$
$504$	0	0
$505$	−3.91552e11	−0.267903
$506$	0	0
$507$	−2.64083e12	−1.77502
$508$	0	0
$509$	2.51799e12	1.66274	0.831370	−	0.555720i	$-0.187557\pi$
$509$	2.51799e12	1.66274	0.831370	+	0.555720i	$0.187557\pi$
$510$	0	0
$511$	−5.40482e11	−0.350661
$512$	0	0
$513$	−1.37656e12	−0.877538
$514$	0	0
$515$	1.23670e12	0.774695
$516$	0	0
$517$	−2.64167e12	−1.62619
$518$	0	0
$519$	−1.25748e12	−0.760759
$520$	0	0
$521$	1.44956e12	0.861920	0.430960	−	0.902371i	$-0.358175\pi$
$521$	1.44956e12	0.861920	0.430960	+	0.902371i	$0.358175\pi$
$522$	0	0
$523$	−9.48490e11	−0.554339	−0.277169	−	0.960821i	$-0.589396\pi$
$523$	−9.48490e11	−0.554339	−0.277169	+	0.960821i	$0.589396\pi$
$524$	0	0
$525$	−1.39882e11	−0.0803609
$526$	0	0
$527$	−8.92505e10	−0.0504038
$528$	0	0
$529$	−1.80006e12	−0.999392
$530$	0	0
$531$	2.69104e11	0.146891
$532$	0	0
$533$	−2.42723e12	−1.30268
$534$	0	0
$535$	5.72672e11	0.302213
$536$	0	0
$537$	−8.13628e11	−0.422223
$538$	0	0
$539$	−4.35550e11	−0.222274
$540$	0	0
$541$	2.01818e12	1.01291	0.506457	−	0.862265i	$-0.330955\pi$
$541$	2.01818e12	1.01291	0.506457	+	0.862265i	$0.330955\pi$
$542$	0	0
$543$	−3.82596e12	−1.88861
$544$	0	0
$545$	1.45714e11	0.0707485
$546$	0	0
$547$	−1.93439e12	−0.923851	−0.461925	−	0.886919i	$-0.652841\pi$
$547$	−1.93439e12	−0.923851	−0.461925	+	0.886919i	$0.652841\pi$
$548$	0	0
$549$	−3.19673e11	−0.150187
$550$	0	0
$551$	−2.66857e12	−1.23338
$552$	0	0
$553$	−1.17347e12	−0.533592
$554$	0	0
$555$	1.61228e12	0.721310
$556$	0	0
$557$	−3.83146e12	−1.68661	−0.843306	−	0.537433i	$-0.819394\pi$
$557$	−3.83146e12	−1.68661	−0.843306	+	0.537433i	$0.819394\pi$
$558$	0	0
$559$	−2.74911e11	−0.119080
$560$	0	0
$561$	3.39055e12	1.44523
$562$	0	0
$563$	1.64076e12	0.688268	0.344134	−	0.938920i	$-0.388172\pi$
$563$	1.64076e12	0.688268	0.344134	+	0.938920i	$0.388172\pi$
$564$	0	0
$565$	−6.05285e11	−0.249886
$566$	0	0
$567$	−1.03549e12	−0.420748
$568$	0	0
$569$	−1.89841e12	−0.759250	−0.379625	−	0.925140i	$-0.623947\pi$
$569$	−1.89841e12	−0.759250	−0.379625	+	0.925140i	$0.623947\pi$
$570$	0	0
$571$	−6.55454e11	−0.258036	−0.129018	−	0.991642i	$-0.541182\pi$
$571$	−6.55454e11	−0.258036	−0.129018	+	0.991642i	$0.541182\pi$
$572$	0	0
$573$	3.91857e12	1.51856
$574$	0	0
$575$	−1.29247e10	−0.00493076
$576$	0	0
$577$	−1.39667e11	−0.0524569	−0.0262285	−	0.999656i	$-0.508350\pi$
$577$	−1.39667e11	−0.0524569	−0.0262285	+	0.999656i	$0.508350\pi$
$578$	0	0
$579$	−1.57714e12	−0.583200
$580$	0	0
$581$	1.54411e12	0.562195
$582$	0	0
$583$	5.09038e12	1.82491
$584$	0	0
$585$	−2.69350e11	−0.0950857
$586$	0	0
$587$	−1.49167e12	−0.518563	−0.259281	−	0.965802i	$-0.583486\pi$
$587$	−1.49167e12	−0.518563	−0.259281	+	0.965802i	$0.583486\pi$
$588$	0	0
$589$	1.59898e11	0.0547423
$590$	0	0
$591$	−9.75982e11	−0.329077
$592$	0	0
$593$	−2.70872e12	−0.899534	−0.449767	−	0.893146i	$-0.648493\pi$
$593$	−2.70872e12	−0.899534	−0.449767	+	0.893146i	$0.648493\pi$
$594$	0	0
$595$	−4.51522e11	−0.147690
$596$	0	0
$597$	−1.36630e12	−0.440210
$598$	0	0
$599$	4.41722e12	1.40194	0.700969	−	0.713192i	$-0.252751\pi$
$599$	4.41722e12	1.40194	0.700969	+	0.713192i	$0.252751\pi$
$600$	0	0
$601$	1.67653e12	0.524173	0.262087	−	0.965044i	$-0.415589\pi$
$601$	1.67653e12	0.524173	0.262087	+	0.965044i	$0.415589\pi$
$602$	0	0
$603$	−7.94260e11	−0.244644
$604$	0	0
$605$	−2.09398e12	−0.635438
$606$	0	0
$607$	−4.52406e12	−1.35263	−0.676315	−	0.736612i	$-0.736424\pi$
$607$	−4.52406e12	−1.35263	−0.676315	+	0.736612i	$0.736424\pi$
$608$	0	0
$609$	−1.77273e12	−0.522234
$610$	0	0
$611$	5.88303e12	1.70772
$612$	0	0
$613$	−4.63979e12	−1.32717	−0.663585	−	0.748101i	$-0.730966\pi$
$613$	−4.63979e12	−1.32717	−0.663585	+	0.748101i	$0.730966\pi$
$614$	0	0
$615$	−1.34469e12	−0.379040
$616$	0	0
$617$	2.10707e11	0.0585322	0.0292661	−	0.999572i	$-0.490683\pi$
$617$	2.10707e11	0.0585322	0.0292661	+	0.999572i	$0.490683\pi$
$618$	0	0
$619$	3.00552e12	0.822833	0.411417	−	0.911447i	$-0.365034\pi$
$619$	3.00552e12	0.822833	0.411417	+	0.911447i	$0.365034\pi$
$620$	0	0
$621$	−8.44919e10	−0.0227983
$622$	0	0
$623$	2.12152e11	0.0564222
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−6.07436e12	−1.56963
$628$	0	0
$629$	5.20424e12	1.32565
$630$	0	0
$631$	−4.21886e12	−1.05941	−0.529704	−	0.848183i	$-0.677697\pi$
$631$	−4.21886e12	−1.05941	−0.529704	+	0.848183i	$0.677697\pi$
$632$	0	0
$633$	−4.02919e12	−0.997473
$634$	0	0
$635$	1.52277e12	0.371665
$636$	0	0
$637$	9.69977e11	0.233418
$638$	0	0
$639$	9.04831e10	0.0214691
$640$	0	0
$641$	1.54927e12	0.362465	0.181232	−	0.983440i	$-0.441991\pi$
$641$	1.54927e12	0.362465	0.181232	+	0.983440i	$0.441991\pi$
$642$	0	0
$643$	5.46397e12	1.26055	0.630274	−	0.776373i	$-0.282943\pi$
$643$	5.46397e12	1.26055	0.630274	+	0.776373i	$0.282943\pi$
$644$	0	0
$645$	−1.52301e11	−0.0346486
$646$	0	0
$647$	−2.39799e12	−0.537994	−0.268997	−	0.963141i	$-0.586692\pi$
$647$	−2.39799e12	−0.537994	−0.268997	+	0.963141i	$0.586692\pi$
$648$	0	0
$649$	−7.93807e12	−1.75636
$650$	0	0
$651$	1.06220e11	0.0231788
$652$	0	0
$653$	3.11297e12	0.669987	0.334993	−	0.942221i	$-0.391266\pi$
$653$	3.11297e12	0.669987	0.334993	+	0.942221i	$0.391266\pi$
$654$	0	0
$655$	8.38863e11	0.178076
$656$	0	0
$657$	−5.76565e11	−0.120727
$658$	0	0
$659$	−4.45743e12	−0.920661	−0.460330	−	0.887748i	$-0.652269\pi$
$659$	−4.45743e12	−0.920661	−0.460330	+	0.887748i	$0.652269\pi$
$660$	0	0
$661$	−1.95869e12	−0.399079	−0.199540	−	0.979890i	$-0.563945\pi$
$661$	−1.95869e12	−0.399079	−0.199540	+	0.979890i	$0.563945\pi$
$662$	0	0
$663$	−7.55079e12	−1.51769
$664$	0	0
$665$	8.08928e11	0.160403
$666$	0	0
$667$	−1.63795e11	−0.0320431
$668$	0	0
$669$	6.73717e12	1.30035
$670$	0	0
$671$	9.42977e12	1.79577
$672$	0	0
$673$	7.14464e12	1.34250	0.671248	−	0.741233i	$-0.265759\pi$
$673$	7.14464e12	1.34250	0.671248	+	0.741233i	$0.265759\pi$
$674$	0	0
$675$	9.97508e11	0.184948
$676$	0	0
$677$	−2.72625e12	−0.498789	−0.249395	−	0.968402i	$-0.580232\pi$
$677$	−2.72625e12	−0.498789	−0.249395	+	0.968402i	$0.580232\pi$
$678$	0	0
$679$	−3.12507e12	−0.564216
$680$	0	0
$681$	−4.47531e11	−0.0797372
$682$	0	0
$683$	−2.20874e12	−0.388376	−0.194188	−	0.980964i	$-0.562207\pi$
$683$	−2.20874e12	−0.388376	−0.194188	+	0.980964i	$0.562207\pi$
$684$	0	0
$685$	−1.79989e12	−0.312348
$686$	0	0
$687$	1.15602e13	1.97998
$688$	0	0
$689$	−1.13364e13	−1.91640
$690$	0	0
$691$	3.29389e12	0.549614	0.274807	−	0.961499i	$-0.411386\pi$
$691$	3.29389e12	0.549614	0.274807	+	0.961499i	$0.411386\pi$
$692$	0	0
$693$	−4.64628e11	−0.0765255
$694$	0	0
$695$	3.01239e12	0.489755
$696$	0	0
$697$	−4.34050e12	−0.696615
$698$	0	0
$699$	−8.23380e12	−1.30453
$700$	0	0
$701$	−4.44986e12	−0.696009	−0.348004	−	0.937493i	$-0.613141\pi$
$701$	−4.44986e12	−0.696009	−0.348004	+	0.937493i	$0.613141\pi$
$702$	0	0
$703$	−9.32370e12	−1.43976
$704$	0	0
$705$	3.25922e12	0.496893
$706$	0	0
$707$	1.50419e12	0.226420
$708$	0	0
$709$	−5.84706e12	−0.869020	−0.434510	−	0.900667i	$-0.643078\pi$
$709$	−5.84706e12	−0.869020	−0.434510	+	0.900667i	$0.643078\pi$
$710$	0	0
$711$	−1.25181e12	−0.183707
$712$	0	0
$713$	9.81439e9	0.00142220
$714$	0	0
$715$	7.94531e12	1.13693
$716$	0	0
$717$	−1.12229e13	−1.58588
$718$	0	0
$719$	−1.53750e12	−0.214553	−0.107276	−	0.994229i	$-0.534213\pi$
$719$	−1.53750e12	−0.214553	−0.107276	+	0.994229i	$0.534213\pi$
$720$	0	0
$721$	−4.75089e12	−0.654737
$722$	0	0
$723$	8.36504e12	1.13853
$724$	0	0
$725$	1.93375e12	0.259944
$726$	0	0
$727$	−1.05931e13	−1.40643	−0.703217	−	0.710976i	$-0.748254\pi$
$727$	−1.05931e13	−1.40643	−0.703217	+	0.710976i	$0.748254\pi$
$728$	0	0
$729$	6.39185e12	0.838210
$730$	0	0
$731$	−4.91610e11	−0.0636785
$732$	0	0
$733$	8.19645e10	0.0104872	0.00524358	−	0.999986i	$-0.498331\pi$
$733$	8.19645e10	0.0104872	0.00524358	+	0.999986i	$0.498331\pi$
$734$	0	0
$735$	5.37370e11	0.0679173
$736$	0	0
$737$	2.34292e13	2.92519
$738$	0	0
$739$	−2.24445e12	−0.276828	−0.138414	−	0.990374i	$-0.544200\pi$
$739$	−2.24445e12	−0.276828	−0.138414	+	0.990374i	$0.544200\pi$
$740$	0	0
$741$	1.35277e13	1.64832
$742$	0	0
$743$	−5.12115e12	−0.616478	−0.308239	−	0.951309i	$-0.599740\pi$
$743$	−5.12115e12	−0.616478	−0.308239	+	0.951309i	$0.599740\pi$
$744$	0	0
$745$	8.46378e11	0.100661
$746$	0	0
$747$	1.64720e12	0.193555
$748$	0	0
$749$	−2.19997e12	−0.255417
$750$	0	0
$751$	3.53215e12	0.405191	0.202595	−	0.979263i	$-0.435062\pi$
$751$	3.53215e12	0.405191	0.202595	+	0.979263i	$0.435062\pi$
$752$	0	0
$753$	9.50202e12	1.07706
$754$	0	0
$755$	2.33919e12	0.262002
$756$	0	0
$757$	−2.53747e12	−0.280847	−0.140423	−	0.990092i	$-0.544846\pi$
$757$	−2.53747e12	−0.280847	−0.140423	+	0.990092i	$0.544846\pi$
$758$	0	0
$759$	−3.72840e11	−0.0407788
$760$	0	0
$761$	1.19773e13	1.29457	0.647287	−	0.762247i	$-0.275904\pi$
$761$	1.19773e13	1.29457	0.647287	+	0.762247i	$0.275904\pi$
$762$	0	0
$763$	−5.59775e11	−0.0597934
$764$	0	0
$765$	−4.81666e11	−0.0508475
$766$	0	0
$767$	1.76782e13	1.84442
$768$	0	0
$769$	1.81034e13	1.86677	0.933385	−	0.358877i	$-0.116840\pi$
$769$	1.81034e13	1.86677	0.933385	+	0.358877i	$0.116840\pi$
$770$	0	0
$771$	1.15184e13	1.17395
$772$	0	0
$773$	−1.85136e13	−1.86501	−0.932507	−	0.361152i	$-0.882384\pi$
$773$	−1.85136e13	−1.86501	−0.932507	+	0.361152i	$0.882384\pi$
$774$	0	0
$775$	−1.15868e11	−0.0115374
$776$	0	0
$777$	−6.19373e12	−0.609618
$778$	0	0
$779$	7.77627e12	0.756577
$780$	0	0
$781$	−2.66908e12	−0.256704
$782$	0	0
$783$	1.26415e13	1.20190
$784$	0	0
$785$	2.72362e12	0.255996
$786$	0	0
$787$	−1.43216e11	−0.0133077	−0.00665386	−	0.999978i	$-0.502118\pi$
$787$	−1.43216e11	−0.0133077	−0.00665386	+	0.999978i	$0.502118\pi$
$788$	0	0
$789$	−1.81635e13	−1.66861
$790$	0	0
$791$	2.32526e12	0.211192
$792$	0	0
$793$	−2.10002e13	−1.88580
$794$	0	0
$795$	−6.28038e12	−0.557614
$796$	0	0
$797$	1.18945e13	1.04420	0.522102	−	0.852883i	$-0.325148\pi$
$797$	1.18945e13	1.04420	0.522102	+	0.852883i	$0.325148\pi$
$798$	0	0
$799$	1.05204e13	0.913209
$800$	0	0
$801$	2.26315e11	0.0194253
$802$	0	0
$803$	1.70076e13	1.44352
$804$	0	0
$805$	4.96514e10	0.00416725
$806$	0	0
$807$	6.04563e12	0.501777
$808$	0	0
$809$	−1.71588e13	−1.40838	−0.704189	−	0.710013i	$-0.748689\pi$
$809$	−1.71588e13	−1.40838	−0.704189	+	0.710013i	$0.748689\pi$
$810$	0	0
$811$	−2.13787e12	−0.173535	−0.0867675	−	0.996229i	$-0.527654\pi$
$811$	−2.13787e12	−0.173535	−0.0867675	+	0.996229i	$0.527654\pi$
$812$	0	0
$813$	−3.74511e12	−0.300647
$814$	0	0
$815$	−6.04014e12	−0.479554
$816$	0	0
$817$	8.80749e11	0.0691596
$818$	0	0
$819$	1.03473e12	0.0803620
$820$	0	0
$821$	1.78524e13	1.37136	0.685681	−	0.727902i	$-0.259504\pi$
$821$	1.78524e13	1.37136	0.685681	+	0.727902i	$0.259504\pi$
$822$	0	0
$823$	1.34474e13	1.02174	0.510870	−	0.859658i	$-0.329323\pi$
$823$	1.34474e13	1.02174	0.510870	+	0.859658i	$0.329323\pi$
$824$	0	0
$825$	4.40173e12	0.330811
$826$	0	0
$827$	1.31321e13	0.976243	0.488121	−	0.872776i	$-0.337682\pi$
$827$	1.31321e13	0.976243	0.488121	+	0.872776i	$0.337682\pi$
$828$	0	0
$829$	1.05599e12	0.0776537	0.0388269	−	0.999246i	$-0.487638\pi$
$829$	1.05599e12	0.0776537	0.0388269	+	0.999246i	$0.487638\pi$
$830$	0	0
$831$	−2.86122e13	−2.08136
$832$	0	0
$833$	1.73457e12	0.124821
$834$	0	0
$835$	6.32487e12	0.450259
$836$	0	0
$837$	−7.57462e11	−0.0533453
$838$	0	0
$839$	5.57545e12	0.388464	0.194232	−	0.980956i	$-0.437778\pi$
$839$	5.57545e12	0.388464	0.194232	+	0.980956i	$0.437778\pi$
$840$	0	0
$841$	9.99942e12	0.689275
$842$	0	0
$843$	1.84026e13	1.25503
$844$	0	0
$845$	−1.10665e13	−0.746716
$846$	0	0
$847$	8.04423e12	0.537043
$848$	0	0
$849$	1.49895e13	0.990154
$850$	0	0
$851$	−5.72282e11	−0.0374048
$852$	0	0
$853$	−3.98443e12	−0.257689	−0.128844	−	0.991665i	$-0.541127\pi$
$853$	−3.98443e12	−0.257689	−0.128844	+	0.991665i	$0.541127\pi$
$854$	0	0
$855$	8.62933e11	0.0552242
$856$	0	0
$857$	−9.79861e12	−0.620513	−0.310257	−	0.950653i	$-0.600415\pi$
$857$	−9.79861e12	−0.620513	−0.310257	+	0.950653i	$0.600415\pi$
$858$	0	0
$859$	−1.46589e13	−0.918615	−0.459307	−	0.888277i	$-0.651902\pi$
$859$	−1.46589e13	−0.918615	−0.459307	+	0.888277i	$0.651902\pi$
$860$	0	0
$861$	5.16577e12	0.320347
$862$	0	0
$863$	1.52373e13	0.935106	0.467553	−	0.883965i	$-0.345136\pi$
$863$	1.52373e13	0.935106	0.467553	+	0.883965i	$0.345136\pi$
$864$	0	0
$865$	−5.26952e12	−0.320036
$866$	0	0
$867$	4.18408e12	0.251486
$868$	0	0
$869$	3.69262e13	2.19657
$870$	0	0
$871$	−5.21772e13	−3.07184
$872$	0	0
$873$	−3.33370e12	−0.194251
$874$	0	0
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	1.91267e13	1.09180	0.545898	−	0.837852i	$-0.316189\pi$
$877$	1.91267e13	1.09180	0.545898	+	0.837852i	$0.316189\pi$
$878$	0	0
$879$	−5.33950e12	−0.301683
$880$	0	0
$881$	−2.59233e13	−1.44977	−0.724885	−	0.688870i	$-0.758107\pi$
$881$	−2.59233e13	−1.44977	−0.724885	+	0.688870i	$0.758107\pi$
$882$	0	0
$883$	−2.57959e13	−1.42800	−0.714000	−	0.700146i	$-0.753118\pi$
$883$	−2.57959e13	−1.42800	−0.714000	+	0.700146i	$0.753118\pi$
$884$	0	0
$885$	9.79378e12	0.536668
$886$	0	0
$887$	−1.37167e13	−0.744036	−0.372018	−	0.928226i	$-0.621334\pi$
$887$	−1.37167e13	−0.744036	−0.372018	+	0.928226i	$0.621334\pi$
$888$	0	0
$889$	−5.84986e12	−0.314114
$890$	0	0
$891$	3.25842e13	1.73204
$892$	0	0
$893$	−1.88478e13	−0.991814
$894$	0	0
$895$	−3.40955e12	−0.177621
$896$	0	0
$897$	8.30320e11	0.0428232
$898$	0	0
$899$	−1.46840e12	−0.0749768
$900$	0	0
$901$	−2.02723e13	−1.02480
$902$	0	0
$903$	5.85081e11	0.0292834
$904$	0	0
$905$	−1.60329e13	−0.794498
$906$	0	0
$907$	−1.35080e13	−0.662764	−0.331382	−	0.943497i	$-0.607515\pi$
$907$	−1.35080e13	−0.662764	−0.331382	+	0.943497i	$0.607515\pi$
$908$	0	0
$909$	1.60461e12	0.0779527
$910$	0	0
$911$	−2.11043e13	−1.01517	−0.507584	−	0.861602i	$-0.669461\pi$
$911$	−2.11043e13	−1.01517	−0.507584	+	0.861602i	$0.669461\pi$
$912$	0	0
$913$	−4.85894e13	−2.31432
$914$	0	0
$915$	−1.16342e13	−0.548708
$916$	0	0
$917$	−3.22258e12	−0.150502
$918$	0	0
$919$	−9.14602e12	−0.422973	−0.211486	−	0.977381i	$-0.567830\pi$
$919$	−9.14602e12	−0.422973	−0.211486	+	0.977381i	$0.567830\pi$
$920$	0	0
$921$	−1.36432e13	−0.624809
$922$	0	0
$923$	5.94409e12	0.269574
$924$	0	0
$925$	6.75633e12	0.303440
$926$	0	0
$927$	−5.06807e12	−0.225415
$928$	0	0
$929$	−3.61138e13	−1.59075	−0.795376	−	0.606116i	$-0.792727\pi$
$929$	−3.61138e13	−1.59075	−0.795376	+	0.606116i	$0.792727\pi$
$930$	0	0
$931$	−3.10758e12	−0.135565
$932$	0	0
$933$	4.00855e13	1.73189
$934$	0	0
$935$	1.42082e13	0.607978
$936$	0	0
$937$	2.24622e13	0.951971	0.475985	−	0.879453i	$-0.342092\pi$
$937$	2.24622e13	0.951971	0.475985	+	0.879453i	$0.342092\pi$
$938$	0	0
$939$	2.65696e12	0.111530
$940$	0	0
$941$	3.69741e12	0.153725	0.0768624	−	0.997042i	$-0.475510\pi$
$941$	3.69741e12	0.153725	0.0768624	+	0.997042i	$0.475510\pi$
$942$	0	0
$943$	4.77302e11	0.0196558
$944$	0	0
$945$	−3.83203e12	−0.156310
$946$	0	0
$947$	−2.26456e13	−0.914976	−0.457488	−	0.889216i	$-0.651251\pi$
$947$	−2.26456e13	−0.914976	−0.457488	+	0.889216i	$0.651251\pi$
$948$	0	0
$949$	−3.78762e13	−1.51589
$950$	0	0
$951$	2.62888e13	1.04222
$952$	0	0
$953$	−1.24447e13	−0.488726	−0.244363	−	0.969684i	$-0.578579\pi$
$953$	−1.24447e13	−0.488726	−0.244363	+	0.969684i	$0.578579\pi$
$954$	0	0
$955$	1.64210e13	0.638827
$956$	0	0
$957$	5.57833e13	2.14981
$958$	0	0
$959$	6.91446e12	0.263982
$960$	0	0
$961$	−2.63516e13	−0.996672
$962$	0	0
$963$	−2.34685e12	−0.0879360
$964$	0	0
$965$	−6.60909e12	−0.245340
$966$	0	0
$967$	−2.20795e13	−0.812026	−0.406013	−	0.913867i	$-0.633081\pi$
$967$	−2.20795e13	−0.812026	−0.406013	+	0.913867i	$0.633081\pi$
$968$	0	0
$969$	2.41910e13	0.881447
$970$	0	0
$971$	−7.55994e12	−0.272918	−0.136459	−	0.990646i	$-0.543572\pi$
$971$	−7.55994e12	−0.272918	−0.136459	+	0.990646i	$0.543572\pi$
$972$	0	0
$973$	−1.15724e13	−0.413919
$974$	0	0
$975$	−9.80271e12	−0.347397
$976$	0	0
$977$	−3.54428e13	−1.24452	−0.622260	−	0.782810i	$-0.713785\pi$
$977$	−3.54428e13	−1.24452	−0.622260	+	0.782810i	$0.713785\pi$
$978$	0	0
$979$	−6.67588e12	−0.232266
$980$	0	0
$981$	−5.97146e11	−0.0205859
$982$	0	0
$983$	4.27895e13	1.46166	0.730831	−	0.682559i	$-0.239133\pi$
$983$	4.27895e13	1.46166	0.730831	+	0.682559i	$0.239133\pi$
$984$	0	0
$985$	−4.08990e12	−0.138436
$986$	0	0
$987$	−1.25206e13	−0.419951
$988$	0	0
$989$	5.40597e10	0.00179676
$990$	0	0
$991$	1.46201e13	0.481526	0.240763	−	0.970584i	$-0.422602\pi$
$991$	1.46201e13	0.481526	0.240763	+	0.970584i	$0.422602\pi$
$992$	0	0
$993$	−3.76775e13	−1.22973
$994$	0	0
$995$	−5.72553e12	−0.185187
$996$	0	0
$997$	−4.88095e13	−1.56450	−0.782251	−	0.622963i	$-0.785929\pi$
$997$	−4.88095e13	−1.56450	−0.782251	+	0.622963i	$0.785929\pi$
$998$	0	0
$999$	4.41679e13	1.40302

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.10.a.c.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.10.a.c.1.1	✓	4	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	140.a (trivial)

\(f(q)\)	\(=\)	\(q-149.145 q^{3} -625.000 q^{5} +2401.00 q^{7} +2561.29 q^{9} +O(q^{10})\)	\(q-149.145 q^{3} -625.000 q^{5} +2401.00 q^{7} +2561.29 q^{9} -75553.4 q^{11} +168258. q^{13} +93215.8 q^{15} +300889. q^{17} -539060. q^{19} -358098. q^{21} -33087.1 q^{23} +390625. q^{25} +2.55362e6 q^{27} +4.95041e6 q^{29} -296623. q^{31} +1.12684e7 q^{33} -1.50062e6 q^{35} +1.72962e7 q^{37} -2.50949e7 q^{39} -1.44256e7 q^{41} -1.63386e6 q^{43} -1.60081e6 q^{45} +3.49642e7 q^{47} +5.76480e6 q^{49} -4.48761e7 q^{51} -6.73746e7 q^{53} +4.72209e7 q^{55} +8.03983e7 q^{57} +1.05066e8 q^{59} -1.24809e8 q^{61} +6.14967e6 q^{63} -1.05162e8 q^{65} -3.10101e8 q^{67} +4.93479e6 q^{69} +3.53271e7 q^{71} -2.25107e8 q^{73} -5.82598e7 q^{75} -1.81404e8 q^{77} -4.88743e8 q^{79} -4.31274e8 q^{81} +6.43113e8 q^{83} -1.88056e8 q^{85} -7.38330e8 q^{87} +8.83597e7 q^{89} +4.03989e8 q^{91} +4.42399e7 q^{93} +3.36913e8 q^{95} -1.30157e9 q^{97} -1.93514e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 43 q^{3} - 2500 q^{5} + 9604 q^{7} + 26935 q^{9}+O(q^{10}) \) 4 * q + 43 * q^3 - 2500 * q^5 + 9604 * q^7 + 26935 * q^9	\( 4 q + 43 q^{3} - 2500 q^{5} + 9604 q^{7} + 26935 q^{9} - 89509 q^{11} - 26951 q^{13} - 26875 q^{15} - 97685 q^{17} + 57614 q^{19} + 103243 q^{21} + 704846 q^{23} + 1562500 q^{25} + 3392209 q^{27} + 6396351 q^{29} - 2177396 q^{31} - 1814697 q^{33} - 6002500 q^{35} - 4832420 q^{37} - 20266753 q^{39} - 33404930 q^{41} - 23422170 q^{43} - 16834375 q^{45} - 30030109 q^{47} + 23059204 q^{49} - 101036823 q^{51} - 29121822 q^{53} + 55943125 q^{55} + 46465094 q^{57} + 187841888 q^{59} - 320985014 q^{61} + 64670935 q^{63} + 16844375 q^{65} - 257915784 q^{67} - 342824850 q^{69} + 95380960 q^{71} - 274354616 q^{73} + 16796875 q^{75} - 214911109 q^{77} - 369481275 q^{79} - 1415449364 q^{81} - 87594732 q^{83} + 61053125 q^{85} - 971702439 q^{87} - 1556249690 q^{89} - 64709351 q^{91} - 1709587660 q^{93} - 36008750 q^{95} - 3692761793 q^{97} - 1570184394 q^{99}+O(q^{100}) \) 4 * q + 43 * q^3 - 2500 * q^5 + 9604 * q^7 + 26935 * q^9 - 89509 * q^11 - 26951 * q^13 - 26875 * q^15 - 97685 * q^17 + 57614 * q^19 + 103243 * q^21 + 704846 * q^23 + 1562500 * q^25 + 3392209 * q^27 + 6396351 * q^29 - 2177396 * q^31 - 1814697 * q^33 - 6002500 * q^35 - 4832420 * q^37 - 20266753 * q^39 - 33404930 * q^41 - 23422170 * q^43 - 16834375 * q^45 - 30030109 * q^47 + 23059204 * q^49 - 101036823 * q^51 - 29121822 * q^53 + 55943125 * q^55 + 46465094 * q^57 + 187841888 * q^59 - 320985014 * q^61 + 64670935 * q^63 + 16844375 * q^65 - 257915784 * q^67 - 342824850 * q^69 + 95380960 * q^71 - 274354616 * q^73 + 16796875 * q^75 - 214911109 * q^77 - 369481275 * q^79 - 1415449364 * q^81 - 87594732 * q^83 + 61053125 * q^85 - 971702439 * q^87 - 1556249690 * q^89 - 64709351 * q^91 - 1709587660 * q^93 - 36008750 * q^95 - 3692761793 * q^97 - 1570184394 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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